Quantum Spin Hall Effect and Topological Phase Transition ... · Generalization of the quantum Hall...
Transcript of Quantum Spin Hall Effect and Topological Phase Transition ... · Generalization of the quantum Hall...
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Quantum Spin Hall Effect AndTopological Phase Transition in
HgTe Quantum Wells
B. Andrei BernevigPrinceton Center for Theoretical Physics
Asilomar, June 2007
Taylor L. Hughes, Shou-Cheng ZhangStanford University
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Generalization of the quantum Hall effect
Fspinkijkspin
i
j ekEJ != "#"
h
e
q
pEJ HjijHi
2
== !"!
• Quantum Hall effect exists in D=2, due to Lorentz force.
• Natural generalization to D=3, due to spin-orbit force:
• 3D hole systems (Murakami et al, Science, PRB)• 2D electron systems (Sinova et al, PRL)
GaAsy
z
x
B
EJ
x: current direction y: spin directionz: electric field
GaAs
E
rx
y
z
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Response to Electric Field
E
Suppose E in y-direction. Then therewill be an spin current flowing in the x-direction with spinspolarized in the z-direction.
xy
z
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Response to Electric Field
Ex
y
z
No net charge flow since the same number of electrons flow in each direction.
Since system has Fermi surfaceit dissipates, has longitudinalresistivity
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How about quantum spin Hall?
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Quantum Hall Effect
B
With Applied Magnetic Field
Without Applied Magnetic Field, with intrinsic T-breaking (magnetic semiconductors)
First proposed model in graphene(Haldane, PRL 61, 2015 (1988));
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Dirac Fermion Revisited
When gapped, a single Dirac fermionbreaks time reversal
TR
No TR
Bulk-Edge correspondence
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• Physical Understanding: Edge states
Quantum Spin Hall Effect
Without Landau Levels, (non-magnetic semiconductorswith Spin-Orbit coupling)
Graphene, Topological Semicondby correlating the Haldane model withspin (Kane and Mele, PRL (2005),Qi, Wu, Zhang PRB (2006));
With Landau Levels correlated with spin through SO coupling(Bernevig and Zhang, PRL (2006));
• (Sheng et al, PRL, (2005);• Kane and Mele PRL, (2005);• Wu, Bernevig and Zhang PRL (2006);• Xu and Moore PRB (2006) …
effectiveB
!"
=zz
n
ne
n
z2
1
!#$
effectiveB
( ) !"!
=zz
m
me
m
z2
1
!#$
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Z2 Topological classification
• Number of edge state PAIRS on each edge must be oddKane and Mele PRL, (2005);
• Single particle backscattering not TR invariant – not allowed• Umklapp relevant for K<½, opens gap for strong interactions.
Wu, Bernevig and Zhang PRL (2006), Xu and Moore, PRB (2006);
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• The edge states of the QSHE is the 1D helical liquid. Opposite spins havethe opposite chirality at the same edge.
• It is different from the 1D chiral liquid(T breaking), and the 1D spinless fermions.
yy SiSieTeT
!! 22==
• T2=1 for spinless fermions and T2=-1for helical liquids.
1 1
1( ) ( )
R L L R
R L L R R L L R
T T T T
T T
! !
" # # "
+ + ! + +
" # # " " # # "
$ =$ $ = !$
$ $ +$ $ = ! $ $ +$ $
• Single particle backscattering is not possible for helical liquids with oddnumber of fermion pairs!
Helical Liquid - Edge Liquid
• A QSHE with even number of electron pairs for one edge is easy to open a TRinvariant gap. For n=2 pairs:
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Quantum Spin Hall Effect
• Goal: realistic proposal of the QSH state of matter in some material
• Do not require the knowledge of full topology of the bands
• Use only (controlled) pertubation theory
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Bulk Band Structure
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CdTe
HgTe
CdTe
E1
H1
normal
HgTe
CdTe CdTe
H1
E1
inverted
Quantum Well Sub-bands
Let us focus on E1(s-wave), H1(p-wave)
bands close to crossing point
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Envelope Functions
Colors indicate non-zero components for each band at k=0
Effective Hamiltonian E1, H1 bands close to crossing point
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Effective 4-band Model
Two copies of Massive Dirac Equation with opposite massesplus an additional kinetic energy term
A,B,C,D,M are numerical parameters that depend on the well thickness
Tunable graphene:
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Continuum Picture
Each diagonal block Hamiltonian has a parity-anomalyOne Dirac point per spin versus two Dirac point in the graphene model.Expect a quantized response from each block of the form:
But this gives fractional quantum-Hall conductance for each block, so whatis missing?
For each block spin (they have opposite masses)
The Skyrmion number changes by +1 for spin up and -1 for spin down as theDirac mass term is tuned across zero. The change is what we need!
E
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Contributions from Brillouin Zone
k
E
µ
Γ
From Γ point we get the contribution:
There is a “spectator fermion”,generically in a higher energy part ofBZ:
2k2
Does this add to the contribution at Γ or cancel it? Cannot knowunless full BZ structure known, not only low-energy.
Look in one of the Dirac blocks, say for spin up
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Physical Picture
Meron incontinuumpicture:
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Change in Spin Hall Conductance
From k.p we cannot determine this since we only know about the Γ point
But the gap closes at the Γ point at some specific quantum wellthickness. We can determine change in Hall conductance only
Γ
(if we include both blocks)
But which side?
[Δ(# of pairs of edge states)] mod 2 =1 QSHE exists on one side of transition
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Effective tight-binding model
Square lattice with 4-orbitals per site:
!+"#+!# ),(,,(,,,,yxyx
ippippss
Consider only the nearest neighbor hoppingintegrals. Mixing matrix elements betweenthe s and the p states must be odd in k.
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Topology of the tight-Binding Model
X X
X
X
X
X X
X
X
X
X
X X
XX
Ferromagnetic FerromagneticSkyrmion Skyrmion
Critical points
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Edge states of the effective tight-binding model
We can analytically find chiral/anti-chiral edge statesolutions which lie in the bulk insulating gap:
edge
(neglecting kinetic energy term)
Trivial Non-trivial
E1
E1H1
H1
Bulk
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X
X
Experimental Signature (2-terminal)
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Smoking gun for the helical edge state: Magneto-Conductance
BgEg !
The crossing of the helical edgestates is protected by the TRsymmetry. TR breaking termsuch as the Zeeman magneticfield causes a singularperturbation and will open up afull insulating gap:
Conductance now takes theactivated form:
kTBgeTf
/)(
!"#
acknowledge conversations with C. Wu
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Conclusions
• QSH state is a new state ofmatter, topologically distinctfrom the conventionalinsulators.
• It is predicted to exist in HgTequantum wells, in the“inverted” regime, with d>6nm.
• Clear experimental signaturespredicted.
• Experimental realizationpossible.
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V
dc
QSHE Transistor
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Conclusions
• QSHE and helical liquid
• QSHE state of matter must exist in HgTe quantum wells
• Lattice calculations indicate existence in “inverted”regime d>6 nm.
• Two clear experimental signatures
• Possibility to create transistor by relatively biasing CdTeand HgTe layers
• 3D topological insulators?Presented at the PITP/SpinAps Asilomar Conference in June 2007 Brought to you by PITP (www.pitp.phas.ubc.ca)
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X
X
Experimental Signature (6-terminal)
A source-drain current j_LR =
=
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Chern Insulator in Magnetic Field(acknowledge conversations with FDM Haldane)
• Interplay between the intrinsic Chern Insulator and the applied B fieldLandau Physics• Valid for magnetic semiconductors with Quantum Anomalous Hall effect• Also valid for Quantum Spin Hall effect in fully polarized limit
•Add B of flux p/q
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Inversion symmetry breaking in zincblend lattices
Inversion breaking term comes in the form:
which couples E1+, H1- and E1-,H1+ states andis a constant in quasi-2d systems
Gap closes at nodes away from k=0, gapreopens at non-zero value of M/2B.In the inverted regime, the helical edge statecrossing is still robust.
, -spin 3/2 matrices
k
E/t
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TM over magnetic unit cell (2 \times 2)
For each k, the elements of the matrix are polynomials inenergy
Hatsugai Method
Edge states BandsPresented at the PITP/SpinAps Asilomar Conference in June 2007 Brought to you by PITP (www.pitp.phas.ubc.ca)
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Hatsugai Method
q=3
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Generalization of Hatsugai Method
Generalization of Harper Equation to matrixHamiltonian
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Hamiltonian Spectrum: Exact Lattice Results
-15 -12.5 -10 -7.5 -5 -2.5 0
-3
-2
-1
0
1
2
3
4
q=30
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-3 -2 -1 1 2 3
-1
-0.5
0.5
1
-2
0
2
-2
0
2
-2
0
2
-2
0
2
Hamiltonian Spectrum
-10 -8 -6 -4 -2 0
-3
-2
-1
0
1
2
3
4
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Edge State Spectrum
q=10,increasinggap, thruthe cherninsulatortransition
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Quantum Anomalous Hall Effect
Magnetic semiconductor with SO coupling (no Landau levels):
General 2×2Hamiltonian
Example
Rashbar Spin-orbital Coupling
Qi et al, 2006
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Hall Conductivity
Quantum Anomalous Hall Effect
Presented at the PITP/SpinAps Asilomar Conference in June 2007 Brought to you by PITP (www.pitp.phas.ubc.ca)
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Helical Liquid - Edge of Quantum Spin Hall
• Single particle backscattering not TR invariant – not allowed
• Bosonize Umklapp, assume . Umklapp relevant for K<½0ug <
• Ising – like. Ordered at T=0 and TR broken; For T >0, Ising disorders.Mass gap with restored TR kills QSHE, only for strong interactions
• A QSHE with even number of electron pairs for one edge is easy toopen a TR invariant gap. For n=2 pairs:
Presented at the PITP/SpinAps Asilomar Conference in June 2007 Brought to you by PITP (www.pitp.phas.ubc.ca)